Shear modulus

Sliding the faces

Put a thick book on a table and push its top cover sideways. The pages slide a little over each other, and the sides tilt. That is shear.

The shear stress is the sideways force over the area of the face it acts on, $\dfrac{F}{A}$. The shear strain is the angle $\theta$ through which the sides tilt. For small angles,

where $\Delta x$ is how far the top moves and $h$ is the height.

The shear modulus

Within the elastic limit, shear stress is in proportion to shear strain. The constant is the shear modulus (or modulus of rigidity):

For metals $G$ is roughly a third of $Y$: it is easier to shear a metal than to stretch it.

You should find that copper moves about $7.5$ times less than lead, and that a block twice as tall shifts twice as far for the same angle.

Watch out. The force here acts along the face, so it is not the force that pushes the face down. The area in $\dfrac{F}{A}$ is the area of the face the force slides.

Why liquids have no shear modulus

A liquid cannot hold a steady sideways stress: it just keeps flowing. So liquids and gases have no shear modulus. They do resist squeezing, which the next lesson is about.

A worked example

A copper block has a square top face $8.0\ \text{cm}$ on a side and is $4.0\ \text{cm}$ tall. Its base is fixed and a sideways force of $3.2 \times 10^{4}\ \text{N}$ drags its top. How far does the top move? ($G = 4.2 \times 10^{10}\ \text{Pa}$.)

1. Stress. $\dfrac{3.2 \times 10^{4}\ \text{N}}{(0.080\ \text{m})^2} = 5.0 \times 10^{6}\ \text{Pa}$.
2. Angle. $\theta = \dfrac{5.0 \times 10^{6}\ \text{Pa}}{4.2 \times 10^{10}\ \text{Pa}} = 1.19 \times 10^{-4}\ \text{rad}$.
3. Shift. $\Delta x = \theta h = 1.19 \times 10^{-4} \times 0.040\ \text{m} = 4.8 \times 10^{-6}\ \text{m}$, under five micrometres.

Think first. The angle is in radians, a pure number, so multiplying by the height gives a length.

Practise with new numbers

Try it in Python

The program works out the copper block of the example.

Remember these

• Shear stress $= \dfrac{F}{A}$, with $F$ along the face; shear strain $= \theta \approx \dfrac{\Delta x}{h}$.
• $G = \dfrac{F/A}{\theta}$; for metals, about $\dfrac{Y}{3}$.
• Liquids and gases have no shear modulus.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. A shear stress of $4.2 \times 10^{6}\ \text{Pa}$ acts on copper. What is the angle of shear?

2. Two blocks of the same metal have the same top face and the same force on it, but one is twice as tall. Compare their angles and their shifts.

3. Why does honey have no shear modulus, though it is thick?

4. Which is easier to do to a steel bar: stretch it by a strain of $10^{-4}$, or shear it through $10^{-4}\ \text{rad}$?